Difference between revisions of "2016 UNCO Math Contest II Problems/Problem 8"
(Created page with "== Problem == == Solution == == See also == {{UNCO Math Contest box|year=2016|n=II|num-b=7|num-a=9}} [[Category:]]") |
(→Solution) |
||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
== Problem == | == Problem == | ||
| + | Tree | ||
| + | |||
| + | |||
| + | Each circle in this tree diagram is to be assigned a value, chosen from a set <math>S</math>, in such a way that along every pathway down the tree, the assigned values never increase. That is, <math>A \ge B, B \ge C, C \ge D, D \ge E</math>, and <math>A, B, C, D, E \in S</math>. (It is permissible for a value in <math>S</math> to appear more than once.) | ||
| + | |||
| + | (a) How many ways can the tree be so numbered, using | ||
| + | only values chosen from the set <math>S = \{1, . . . , 6\}</math>? | ||
| + | |||
| + | (b) Generalize to the case in which <math>S = \{1, . . . , n\}</math>. Find a formula for the number of ways the | ||
| + | tree can be numbered. | ||
| + | |||
| + | For maximal credit, express your answer in closed form as an explicit algebraic expression in <math>n</math>. | ||
== Solution == | == Solution == | ||
| + | a)<math>994</math> b) <math>\frac{1}{120}n(n + 1)(n + 2)(8n^22 + 11n + 1)</math> | ||
== See also == | == See also == | ||
{{UNCO Math Contest box|year=2016|n=II|num-b=7|num-a=9}} | {{UNCO Math Contest box|year=2016|n=II|num-b=7|num-a=9}} | ||
| − | [[Category:]] | + | [[Category: Intermediate Combinatorics Problems]] |
Latest revision as of 03:03, 13 January 2019
Problem
Tree
Each circle in this tree diagram is to be assigned a value, chosen from a set 
, in such a way that along every pathway down the tree, the assigned values never increase. That is, 
, and 
. (It is permissible for a value in to appear more than once.)
(a) How many ways can the tree be so numbered, using
only values chosen from the set 
?
(b) Generalize to the case in which 
. Find a formula for the number of ways the
tree can be numbered.
For maximal credit, express your answer in closed form as an explicit algebraic expression in 
.
Solution
a)
b) 
See also
| 2016 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
